# Every p-subgroup is well-placed in some Sylow subgroup

## Contents

## Statement

### Statement that involves choosing a Sylow subgroup

Suppose is a finite group, is a prime number, and is a Conjugacy functor (?) on the -subgroups of . Suppose is a -subgroup of . Then, there exists a -Sylow subgroup of such that is a Well-placed subgroup (?) in relative to the conjugacy functor .

### Statement that involves conjugating the subgroup

Suppose is a finite group, is a prime number, and is a Conjugacy functor (?) on the -subgroups of . Suppose is a -subgroup of contained in a -Sylow subgroup . Then, there exists a subgroup of , that is well-placed in , and is conjugate to in .

## Related facts

## Facts used

- Conjugacy functor sends every subgroup to a normalizer-relatively normal subgroup: For any -subgroup , .
- Sylow subgroups exist
- Sylow implies order-dominating

## Proof

**Given**: A finite group , a prime , a conjugacy functor on the -subgroups. A -subgroup of .

**To prove**: There exists a -Sylow subgroup of such that is well-placed in relative to .

**Proof**: Define the following ascending chain of subgroups . , and is a -Sylow subgroup of . By fact (1), this chain of subgroups is an ascending chain of -subgroups. Let be a -Sylow subgroup of containing the union of the s (such a exists by facts (2) and (3)). We now prove that is well-placed in . This follows easily from the definition.